Time series and Cointegrating vectors

I was studying time series from here.
at page 4, I came across this statement:
If $\displaystyle x_{t}$ is a $\displaystyle p$-dimensional vector of non-stationary variables that are $\displaystyle I(1)$, then there could be at most $\displaystyle p-1$ cointegrating vectors.

I know that $\displaystyle p$ vectors of dimension $\displaystyle p$ could be linearly independent.

Can somebody explain me why there could be at most $\displaystyle n-1$ cointegrating vectors?